Linear instability of periodic orbits of free period Lagrangian systems

Alessandro Portaluri; Li Wu; Ran Yang; Alessandro Portaluri; Li Wu; Ran Yang

doi:10.3934/era.2022144

Electronic Research Archive

2022, Volume 30, Issue 8: 2833-2859. doi: 10.3934/era.2022144

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Linear instability of periodic orbits of free period Lagrangian systems

1.
DISAFA, Università degli Studi di Torino, Largo Paolo Braccini 2, 10095 Grugliasco, Torino, Italy
2.
Department of Mathematics, Shandong University, Jinan, Shandong, 250100, China
3.
School of Science, East China University of Technology, Nanchang, Jiangxi, 330013, China

Received: 24 September 2021 Revised: 01 April 2022 Accepted: 01 April 2022 Published: 24 May 2022

Minimizing closed geodesics on surfaces are linearly unstable. By starting with this classical Poincaré's instability result, in the present paper we prove a result that allows to deduce the linear instability of periodic solutions of autonomous Lagrangian systems admitting an orbit cylinder (condition which is satisfied for instance if the periodic orbit is transversally non-degenerate) in terms of the parity properties of a suitable quantity which is obtained by adding the dimension of the configuration space to a suitably defined spectral index. Such a spectral index coincides with the Morse index of the periodic orbit seen as a critical point of the free period action functional in the case the Lagrangian is Tonelli, namely fibrewise strictly convex and superlinear, and it encodes the functional and symplectic properties of the problem.

The main result of the paper is a generalization of the celebrated Poincaré 's instability result for closed geodesics on surfaces and at the same time extends to the autonomous case several previous results which have been proved by the authors (as well as by others) in the case of non-autonomous Lagrangian systems.
- periodic orbits,
- free period Lagrangian systems,
- linear instability,
- Maslov index,
- spectral flow
Citation: Alessandro Portaluri, Li Wu, Ran Yang. Linear instability of periodic orbits of free period Lagrangian systems[J]. Electronic Research Archive, 2022, 30(8): 2833-2859. doi: 10.3934/era.2022144

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Abstract

Minimizing closed geodesics on surfaces are linearly unstable. By starting with this classical Poincaré's instability result, in the present paper we prove a result that allows to deduce the linear instability of periodic solutions of autonomous Lagrangian systems admitting an orbit cylinder (condition which is satisfied for instance if the periodic orbit is transversally non-degenerate) in terms of the parity properties of a suitable quantity which is obtained by adding the dimension of the configuration space to a suitably defined spectral index. Such a spectral index coincides with the Morse index of the periodic orbit seen as a critical point of the free period action functional in the case the Lagrangian is Tonelli, namely fibrewise strictly convex and superlinear, and it encodes the functional and symplectic properties of the problem.

The main result of the paper is a generalization of the celebrated Poincaré 's instability result for closed geodesics on surfaces and at the same time extends to the autonomous case several previous results which have been proved by the authors (as well as by others) in the case of non-autonomous Lagrangian systems.

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